Maxwell Equal Area Law Research Articles

Maxwell's equal area law for Vaidya-Bonner-de Sitter black hole under Lorentz invariance violation

In this study, we investigate the tunneling of fermions with arbitrary spin near the event horizon of a nonstationary Vaidya-Bonner-de Sitter (VBdS) black hole under Lorentz invariance violation (LIV). The modified Hawking temperature of VBdS black holes is calculated by using tortoise coordinate transformation, Feynman prescription, and Wentzel–Kramers–Brillouin approximation. By considering the cosmological constant as a thermodynamic pressure in the extended phase space, we construct a Maxwell's equal area law under LIV and study the phase transitions of VBdS black hole in , , and planes. The LIV increases the length of the liquid-gas coexistence region. The thermodynamic quantities such as the entropy, heat capacity, Helmholtz free energy, internal energy, enthalpy, and Gibbs free energy of the VBdS black hole are discussed. These quantities tend to increase under LIV. The stability of the black hole is also discussed in the presence of LIV.

Joule Thomson expansion, Maxwell equal area law and topological interpretation of Phantom RN AdS black holes

This paper is devoted to study the thermodynamic topological defects, Joule–Thomson (J−T) and Maxwell’s equal area law of Phantom RN AdS black holes (BHs). The inversion temperatures and inversion curves are obtained and by using the isenthalpic curves in the temperature–pressure (T−P) plane to locate the cooling and heating regions. Using Maxwell’s equal-area law, we select different independent conjugate variables to study the phase transitions of Phantom RN AdS BHs. We find that phase transition rely on the electric potential and horizon radius of the BH when its charge is constant. Phase transition of Phantom RN AdS BH are proportional to the ratio of event horizon to its cosmological constant, where the latter is assumed to be constant. Moreover, we consider the Phantom RN AdS BH as defects with a topological nature within thermodynamic domain and examine the local and global topology by computing the winding numbers at the defects, which concludes that the overall topological charge is either equal to 0 or 1.

Recovery of consistency in thermodynamics of regular black holes in Einstein's gravity coupled with nonlinear electrodynamics

As one of candidate theories in the construction of regular black holes, Einstein's gravity coupled with nonlinear electrodynamics has been a topic of great concerns. Owing to the coupling between Einstein's gravity and nonlinear electromagnetic fields, we need to reconsider the first law of thermodynamics, which will lead to a new thermodynamic phase space. In such a phase space, the equation of state accurately describes the complete phase transition process of regular black holes. The Maxwell equal area law strictly holds when the phase transition occurs, and the entropy obeys the Bekenstein-Hawking area formula, which is compatible with the situation in Einstein's gravity.

Generalized Maxwell equal area law and black holes in complex free energy

Maxwell equal area law is an important and traditional analytical tool in thermodynamic phase transition, especially in the calculation of gas-liquid phase transition, which reconciles the theoretical calculation with the experimental results. Undoubtedly, its importance is also self-evident for the black hole thermodynamic system. In this study, we construct a generalized Maxwell equal area law, which allows different states of thermodynamic systems to be within the generalized free energy. The black hole thermodynamic characteristics are spontaneously emerged in the free energy landscape. Furthermore, by analytic continuation, we utilize the properties of analytical functions to investigate some universal characteristics of thermodynamic phase transitions in black holes, and preliminarily establish the counterpart of thermodynamic phase transitions in the complex domain.

Continuous phase transition of the de Sitter spacetime with charged black holes and cloud of strings and quintessence* *Supported by the Natural Science Foundation of China (11705106, 12075143), the Scientific Innovation Foundation of the Higher Education Institutions of Shanxi Province (2020L0471, 2020L0472), the Science Technology Plan Project of Datong City, China (2020153), the Science Foundation of Shanxi Datong University (2022Q1,

Recently, some meaningful results have been obtained by studying the phase transition, critical exponents, and other thermodynamical properties of different black holes. Especially for the Anti-de Sitter (AdS) black holes, their thermodynamical properties nearby the critical point have attracted considerable attention. However, there exists little work on the thermodynamic properties of the de Sitter (dS) spacetime with black holes. In this paper, based on the effective thermodynamical quantities and the method of the Maxwell's equal-area law, we explore the phase equilibrium for the de Sitter spacetime with the charged black holes and the cloud of string and quintessence (i.e., C-dSSQ spacetime). The boundaries of the two-phase coexistence region in both and diagrams are obtained. The coexistent curve and the latent heat of phase transition for this system are also investigated. Furthermore, we analyze the effect of parameters (the state parameter ω and the ratio of two horizon radii /) on the two-phase coexistence region boundary. The results indicate that the phase transition in C-dSSQ spacetime is analogous to that in a van der Waals fluid (vdw) system, which is determined by the electrical potential at the horizon. These results are helpful for understanding the basic properties of black holes and are also of great value for the establishment of quantum gravity.

On heat properties of charged AdS black holes in Gauss-Bonnet gravity coupled with nonlinear electrodynamics

We investigate the heat properties of charged AdS black holes in the Gauss-Bonnet gravity coupled with nonlinear electrodynamics. We consider the thermodynamics of black holes from the perspective of heat capacity and show that the nonlinear electrodynamics can be helpful to improve the thermodynamic stability of black holes. We perform a two-dimensional description in order to reproduce the Hawking temperature, which confirms that the Hawking temperature has an intrinsic topological nature and holds for a higher dimensional spherically symmetric spacetime. We also analyze the Maxwell equal area law and coexistence curve, and find the existence of van der Waals-like phase transitions based on critical exponents. Moreover, we deal with a charged AdS black hole in the Gauss-Bonnet gravity coupled with nonlinear electrodynamics as a working material to study holographic heat engines and obtain an exact expression for the efficiency of a rectangular engine cycle. We then discuss the effects of nonlinear electrodynamics and Gauss-Bonnet couplings on the rectangular engine cycle and compare the efficiency of this cycle with that of the Carnot cycle.

Criticality and phase transition of Kerr–anti-de Sitter black hole with quintessence and cloud of strings

In this paper, a thermodynamical study of the Kerr–anti-de Sitter (KAdS) black hole (BH) encircled by quintessence matter and the cloud of strings is presented. We check the effects of the cloud parameter on the thermodynamical variables including Hawking temperature, entropy, Helmholtz free energy (HFE), Gibbs free energy (GFE), and specific heat capacity. These thermodynamic variables hold the Smarr–Gibbs–Dehum (SGD) relation in the appearance of quintessence and the cloud of the string parameter. Then we calculate the critical expressions of the developed variables in two different ways that are “Van-der-Waals (vdW)-type equation of state” (EoS) and the “Maxwell equal-area law” (MEAL). The latter technique is found to be more reasonable to examine the criticality of the complex BHs. Using MEAL, we plot the phase diagram in a temperature–entropy plane and obtain an isobar, which represents the co-existence region of liquid–gas phases. It is concluded that the uncharged BHs exhibit the same phase transition (PT) to vdW fluid as the temperature drops down its critical value. Also, we discuss the influence of thermal fluctuations (TF) on the stability of KAdS BH.

Phase transition and quantum corrections of quintessential Kerr–Newman black hole with cloud of strings

The purpose of this manuscript is to inspecting the phase transition and the critical phenomenon of charged quintessential Kerr-Newman-anti-de Sitter black hole with cloud of strings. In this regard, we evaluate the thermodynamic factors (Hawking temperature, angular momentum and entropy) in the framework of extended phase space. These factors fulfill the Smarr-Gibbs-Dehum equation in the existence of quintessence and cloud of strings. The critical behavior of thermodynamic factors is discussed via two techniques such as Maxwell equal-area law and Van-der Waal like equation of state . It is obtained that the former one is more effective to observe the critical nature of the complex black holes. Further, we study the phase diagram in T − S plane using equal-area law and obtain an isobar depicting the coexistence era of two phases. It is concluded that black holes lower than the critical temperature exhibit the phase transition same as that of Van-der Waal fluid. At the end, we analyze the influence of thermal variations on the black hole’s stability. • We inspect the critical phenomenon of quintessential string Kerr-Newman-AdS BH. • The critical behavior via Maxwell equal-area law and Van-der Waal EoS is studied. • Below the critical temperature, BHs show the VdW fluid like phase transition. • At the end, we analyze the influence of thermal variations on the black hole’s stability.

Emergent phase, thermodynamic geometry, and criticality of charged black holes from Rényi statistics

Recently, a novel emergent phase can occur from thermodynamic consideration of the asymptotically flat Reissner-Nordstr\"om black hole (RN-AF) using R\'enyi statistics. We present an analysis of the thermodynamical and mechanical stabilities of the RN-AF in both the Gibbs-Boltzmann (GB) and the alternative R\'enyi statistics when charge $q$ and electrostatic potential $\phi$ are treated as pressure and volume, respectively. Interestingly, the emergent phase of the RN-AF can be both thermodynamically and mechanically stable in some range of parameters in the framework of R\'enyi thermodynamics. With the construction of the Maxwell equal area law in $q-\phi$ plane, the coexistence line between the near-extremal black hole phase and the emergent phase can be found in some values of charge which can be associated as the vapor pressure at which the liquid and gas phases coexist. In the aspect of thermodynamic geometry, the microscopic interaction between the black hole microstructures can be repulsive in the R\'enyi description. This implies that a novel correlation between the microstates of a self-gravitating system could be emerged via the nonextensive nature of long-range interaction systems. Finally, we also investigate the critical phenomena of the RN-AF in R\'enyi statistics compared to that of the van der Waals (vdW) fluid and find that the critical exponents of the relevant physical quantities of both systems are identical. This implies that both systems are in the same universality class of the phase transition.

Phase equilibrium and microstructure of topological AdS black holes in massive gravity * *Supported in part by the National Natural Science Foundation of China (11705106, 11705107, 12075143), Science and Technology Innovation Project of Shanxi Higher Education (2016173) and the Doctoral Sustentation Fund of Shanxi Datong University

In order to clearly understand the gravitational theory through the thermal properties of the black hole, it is important to further investigate the first-order phase transition of black holes. In this paper, we adopt different conjugate variables ( , , , and ) and apply Maxwell's equal-area law to study the phase equilibrium of a topological black hole in massive gravity. The condition and latent heat of phase transition are displayed as well as the coexistent curve of . The result shows that the phase transition of this system is the high/low electric potentials one, not only the large/small black holes one. We also analyze the effect of the model's parameters on phase transition. Furthermore we introduce a new order parameter to probe the microstructure of this system. This work will provide the theoretical basis to study the phase structure of topological black holes in massive gravity and to further explore the gravitational theory.

Dynamic property of phase transition for non-linear charged anti-de Sitter black holes * *Supported by the National Natural Science Foundation of China (11705106, 11475107, 12075143), the Natural Science Foundation of Shanxi Province, China (201901D111315), the Natural Science Foundation for Young Scientists of Shanxi Province, China (201901D211441), the Scientific Innovation Foundation of the Higher Education Institutions of Shanxi Province (2020L0471, 2020L0472,

Understanding the thermodynamic phase transition of black holes can provide deep insights into the fundamental properties of black hole gravity and help to establish quantum gravity. In this work, we investigate the phase transition and its dynamics for the charged EPYM AdS black hole. Through reconstructing Maxwell's equal-area law, we find there exists a high-/low-potential black hole (HPBH/LPBL) phase transition, not only the pure large/small black hole phase transition. The Gibbs free energy landscape ( ) is treated as a function of the black hole horizon, which is the order parameter of the phase transition due to thermal fluctuation. From the viewpoint of , the stable HPBH/LPBL states correspond to two wells of , which have the same depth. The unstable intermediate-potential black hole state corresponds to the local maximum of . Then we focus on the probability evolution governed by the Fokker–Planck equation. Through solving the Fokker–Planck equation with different reflection/absorption boundary conditions and initial conditions, the dynamics of switching between the coexistent HPBH and LPBL phases is probed within the first passage time. Furthermore, the effect of temperature on the dynamic properties of the phase transition is also investigated.

Black hole solution and thermal properties in 4D AdS Gauss–Bonnet massive gravity

We consider an Einstein-Gauss-Bonnet massive gravity model in $4D$ AdS spacetime to obtain a possible black hole solution and discuss the horizon structure of this black hole. The real roots of the vanishing metric function lead to various types of horizons. Furthermore, we derive various thermodynamic quantities, thus insuring the validity of the first-law of thermodynamics and the Smarr relation. The thermodynamic quantities are modified in the presence of the massive gravity parameter and also discuss the stability of the system from the heat capacity. Black hole space times can not only possess standard thermodynamics, but also possess phase structures when the cosmological constant is treated as the thermodynamic pressure. The effects of massive parameter and GB parameter on phase transition are also discussed. We also examine the phase structure through Maxwell's equal area law. The first order phase transition occurs only when pressure is lower than its critical value. At critical pressure, the first order phase transition terminates, and the second order phase transition occurs. No phase transition occurs when pressure is greater than its critical value.

Phase transition of non-linear charged Anti-de Sitter black holes * *Supported by the Natural Science Foundation of China (11705106, 11475108, 12075143), the Natural Science Foundation of Shanxi Province, China (201901D111315), the Natural Science Foundation for Young Scientists of Shanxi Province, China (201901D211441), the Scientific Innovation Foundation of the Higher Education Institutions of Shanxi Province (2020L0471, 2020L0472), and the Science Technology

Understanding the thermodynamic phase transition of black holes can provide a deep insight into the fundamental properties of black hole gravity to establish the theory of quantum gravity. We investigate the condition and latent heat of phase transition for non-linear charged AdS black holes using Maxwell's equal-area law. In addition, we analyze the boundary and curve of the two-phase coexistence area in the expanded phase space. We suggest that the phase transition of the non-linear charged AdS black hole with the fixed temperature ( ) is related to the electric potential at the horizon, not only to the location of black hole horizon. Recently, the molecular number density was introduced to study the phase transition and microstructure of black holes. On this basis, we discuss the continuous phase transition of a non-linear charged AdS black hole to reveal the potential microstructure of a black hole by introducing the order parameter and using the scalar curvature.

$$P-V$$ criticality and phase transition of the Kerr-Sen-AdS Black Hole

In this paper, we analyze the critical phenomena and phase transition of the Kerr-Sen-anti-de Sitter black hole in the framework of Maxwell equal-area law. First, we review the exact expressions of thermodynamic quantities such as entropy, Hawking temperature and angular momentum in the extended phase space. These quantities satisfy the usual first law of thermodynamics as well as Smarr–Gibbs–Dehum relation. The critical behavior of thermodynamic quantities is analyzed through two techniques, i.e., Maxwell equal-area law and van der Waal-like equation of state. It is found that the former technique is more proficient to study the critical behavior of complicated black holes. Second, we construct phase diagram in $$T-S$$ conjugate variables and find an isobar using equal-area law. It is concluded that below the critical pressure, black holes observe similar phase transition as that of liquid-gas van der Waal fluid. Finally, we analyze the impact of thermal fluctuations on the stability of considered black hole solution.

Phase transition and thermal fluctuations of quintessential Kerr–Newman–AdS black hole

This paper is devoted to analyzing the critical phenomenon and phase transition of quintessential Kerr–Newman-anti-de Sitter black hole in the framework of Maxwell equal-area law. For this purpose, we first derive thermodynamic quantities such as Hawking temperature, entropy and angular momentum in the context of extended phase space. These quantities satisfy Smarr–Gibbs–Dehum relation in the presence of quintessence matter. We then discuss the critical behavior of thermodynamic quantities through two approaches, i.e., van der Waals-like equation of state and Maxwell equal-area law. It is found that the latter approach is more effective to analyze the critical behavior of the complicated black holes. Using equal-area law, we also study phase diagram in T – S plane and find an isobar which shows the coexistence region of two phases. We conclude that below the critical temperature, black holes show a similar phase transition as that of van der Waals fluid. Finally, we study the effects of thermal fluctuations on the stability of this black hole.

Continuous phase transition and microstructure of charged AdS black hole with quintessence

Previously, the Maxwell equal-area law has been used to discuss the conditions satisfied by the phase transition of charged AdS black holes with cloud of string and quintessence, and it was concluded that black holes have phase transition similar to that of vdW system. The phase transition depends on the electric potential of the black hole and is not the one between a large black hole and a small black hole. On the basis of this result, we study the relation between the latent heat of the phase transition and the parameter of dark energy, and use the Landau continuous phase transition theory to discuss the critical phenomenon of the black hole with quintessence and give the critical exponent. By introducing the number density of the black hole molecules, some properties of the microstructure of black holes are studied in terms of a phase transition. It is found that the electric charge of the black hole and the normalization parameter related to the density of quintessence field play a key role in the phase transition. By constructing the binary fluid model of the black hole molecules, we also discuss the microstructure of charged AdS black holes with a cloud of strings and quintessence.

Maxwell's equal-area law with several pairs of conjugate variables for RN-AdS black holes

In this paper, using Maxwell's equal-area law we study the phase transition of charged AdS black holes by choosing different independent conjugate variables. As is well known, the phase transition can be characterized by the state function of the system, the determination of the phase transition point has nothing to do with the choice of independent conjugate variables. To studying the thermodynamic properties of AdS black holes we give the conditions under which the independent conjugate variables are chosen. When the charge of the black hole is invariable, according to the conditions we find that the phase transition is related to the electric potential and the horizon radius of the charged black hole. Keeping the cosmological constant as a fixed parameter, the phase transition of a charged AdS black hole is related to the ratio of the event horizon to cosmological constant of the black hole. This conclusion is of great importance for us to study the critical phenomenon of black holes and to improve the self-consistent geometry theory of black hole thermodynamic.

New Phase Transition Related to the Black Hole’s Topological Charge

The topological charge ϵ of AdS black hole is introduced by Tian et al. in their papers, where a complete thermodynamic first law is obtained. In this paper, we investigate a new phase transition related to the topological charge in Einstein-Maxwell theory. Firstly, we derive the explicit solutions corresponding to the divergence of specific heat Cϵ and determine the phase transition critical point. Secondly, the T-r curve and T-S curve are investigated and they exhibit an interesting van der Waals system’s behavior. Critical physical quantities are also obtained which are consistent with those derived from the specific heat analysis. Thirdly, a van der Waals system’s swallow tail behavior is observed when ϵ>ϵc in the F-T graph. What is more, the analytic phase transition coexistence lines are obtained by using the Maxwell equal area law and free energy analysis, the results of which are consistent with each other.

Phase Transition of RN-AdS Black Hole with Fixed Electric Charge and Topological Charge

Phase transition of RN-AdS black hole is investigated from a new perspective. Not only is the cosmological constant treated as pressure but also the spatial curvature of black hole is treated as topological charge ϵ. We obtain the extended thermodynamic first law from which the mass is naturally viewed as enthalpy rather than internal energy. In canonical ensemble with fixed topological charge and electric charge Q, interesting van der Waals like oscillatory behavior in T-S and P-V graphs and swallow tail behavior in G-T and G-P graphs is observed. By applying the Maxwell equal area law and analysing the Gibbs free energy, we obtain analytical phase transition coexistence curves which are consistent with each other. The phase diagram is four dimensional with T,P,Q,ϵ.

Parametric phase transition for a Gauss-Bonnet AdS black hole

With the help of the parametric solution of the Maxwell equal area law for the Gauss-Bonnet AdS black hole in five dimensions, we find the second analytical solution to the first order phase transition. We analyze the asymptotic behaviors of some characteristic thermodynamic properties for the small and large black holes at the critical and zero temperatures and also calculate the critical exponents and the corresponding critical amplitudes in detail. Moreover, we give the general form of the thermodynamic scalar curvature based on the Ruppeiner geometry and point out that the attractive interaction dominates in both the small and large black hole phases when the first order phase transition occurs in the five dimensional Gauss-Bonnet AdS black hole.